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# Properties of Cross Product

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#### Properties of Cross Product - Lesson Summary

Property 1: p â†’   x  q â†’   is also a vector.

The cross product of p â†’   and  q â†’   is given by:

p â†’   x q â†’   = | p â†’  | | q â†’  |  sin Ï´ n ^

Hence, the cross product of two vectors p and q, and the unit vector, n, are collinear.

Property 2: p â†’   x  q â†’   = 0 â†’   â‡” p â†’     ||   q â†’   .

Let p â†’   and q â†’   be two non-zero vectors.

If p â†’   âˆ¥  q â†’  , Ï´ = 0o

â‡’ p â†’   x q â†’   = | p â†’  | | q â†’  |  sin 0  n ^   =  0 â†’

(sin 0o = 0)

Observation 1: p â†’   x p â†’   =  0 â†’

Since Ï´ = 0o

Observation 2: p â†’   x ( - p â†’  ) =  0 â†’

Since Ï´ = 180o

Observation 3:

i ^ x i ^ =  0 â†’

j ^ x j ^ = 0 â†’

k ^ x k ^ = 0 â†’

If p â†’   Á q â†’  , Ï´ = 90o

Property 3: Given p â†’   Á q â†’  ,

p â†’   x q â†’   =  | p â†’  | | q â†’  |  sin 90 n ^ =  | p â†’  | | q â†’  |  n ^

â‡’  p â†’    x   q â†’   =  | p â†’  | | q â†’  | | n ^ |  = | p â†’  | | q â†’  |

(sin 90o = 1 and n ^ is a unit vector)

Property 4: Angle Ï´ between two vectors p â†’   and q â†’   is given by:

sin Ï´ = | p _ x q _ | | p _ | | q _ |

p â†’   x q â†’   = | p â†’  | | q â†’  | sin Ï´ n ^ ...(1)

q â†’   x p â†’   =  | p â†’  | | q â†’  | sin Ï´ n ^ , ...(2)

n ^ = - n ^

q â†’   x p â†’   = - | p â†’  | | q â†’  | sin Ï´ n ^ ...(3)

From (1) and (3):

p â†’   x q â†’   = - q â†’    x p â†’

Property 5: The cross product of vectors is not commutative, since p â†’   x q â†’   = - q â†’    x p â†’  .

For mutually perpendicular vectors i ^ and j ^ :

i ^ x j ^ = | i ^ | | j ^ | sin 90 n ^

â‡’  i ^ x j ^ = n ^ (since  | i ^ | = | j ^ | = sin 90 = 1)

â‡’  i ^    x j ^ =  k ^ (since n ^  âŠ¥ i ^    x j ^ )

Since cross product is not commutative:

j ^ x i ^ = -  k ^

Similarly:

j ^ x  k ^ =  i ^ and k ^ x  j ^ = -  i ^

k ^ x i ^ =  j ^ and i ^ x  k ^ = -  j ^

Property 6: If p â†’   and q â†’   represent the adjacent sides of a triangle, then the area of the triangle = ½ |  p â†’   x  q â†’  |.

In âˆ†OPQ, let QR Á OP.

Area of âˆ†OPQ = ½ Base x Height

â‡’ Area of âˆ†OPQ = ½ (OP). ( QR) ...(1)

In right-angled âˆ†OQR:

sin Ï´ = QR OQ

â‡’ QR = OQ sin Ï´

â‡’ Area of âˆ†OPQ = ½ (OP) (OQ) sin Ï´

OP = | p â†’  | and OQ = | q â†’  |

â‡’ Area of âˆ†OPQ = ½ | p â†’  | | q â†’  | sin Ï´

Since |  p â†’   x  q â†’  | = | p â†’  | | q â†’  | sin Ï´

Area of âˆ†OPQ = ½ | p â†’   x q â†’  |

The cross product of p â†’   and q â†’   is given by: p â†’   x q â†’   = | p â†’   | | q â†’  | sin Ï´ n ^

Property 7: If p â†’   and q â†’   represent adjacent sides of a parallelogram, the area of the parallelogram = | p â†’   x q â†’  |.

In parallelogram OPRQ, let QS Á OP.

Area of parallelogram OPRQ = Base x Height

â‡’ Area of parallelogram OPRQ = OP .QS ...(1)

In right-angled âˆ†OQS:

sin Ï´ = QS OQ

â‡’ QS = OQ sin Ï´

â‡’ Area of parallelogram OPRQ = OP . OQ sin Ï´

OP = | p â†’  | and OQ = | q â†’  |

â‡’ Area of parallelogram OPRQ =  | p â†’  | | q â†’  |  sin Ï´

Since | p â†’   x q â†’  | = | p â†’  | | q â†’  | sin Ï´

Area of parallelogram OPRQ = | p â†’   x q â†’  |

Property 7: Distributive property of cross product.

For any three vectors  p â†’   , q â†’   and r â†’    :

p â†’   x ( q â†’    + r â†’  ) =   p â†’   x   q â†’   +   p â†’   x r â†’

Property 8: For any two vectors p â†’   and q â†’   and a scalar Î» :

Î» ( p â†’   x q â†’  ) = (Î»  p â†’  ) x  q â†’   = p â†’   x (Î» q â†’  )

a â†’   =  a1 i ^ + a2  j ^ + a3  k ^

b â†’   =  b1 i ^ + b2  j ^ + b3  k ^

a â†’   x b â†’   = ( a1 i ^ + a2  j ^ + a3  k ^   ) x (b1 i ^ + b2  j ^ + b3  k ^ )

â‡’  a â†’   x b â†’   =  a1 i ^   x (b1 i ^ + b2  j ^ + b3  k ^ ) +   a2  j ^   x (b1 i ^ + b2  j ^ + b3  k ^ ) + a3  k ^   x (b1 i ^ + b2  j ^ + b3  k ^ )

= a1b1 ( i ^   x  i ^ ) + a1b2 ( i ^   x  j ^ ) + a1b3 ( i ^   x  k ^ ) + a2b1 ( j ^   x  i ^ ) + a2b2 ( j ^   x  j ^ ) + a2b3 ( j ^   x  k ^ ) + a3b1 ( k ^   x  i ^ ) + a3b2 ( k ^   x  j ^ ) + a3b3 ( k ^   x  k ^ )

We know:

i ^   x  i ^   =  j ^   x  j ^    =   k ^   x  k ^    = 0

i ^ x j ^ = k ^ and j ^ x i ^ = - k ^

j ^ x k ^ = i ^ and k ^ x j ^ = -  i ^

k ^ x i ^ = j ^ and i ^ x k ^ = - j ^

â‡’  a â†’   x b â†’   =  a1b2 ( i ^   x  j ^ ) + a1b3 ( i ^   x  k ^ ) + a2b1 ( j ^   x  i ^ )  + a2b3 ( j ^   x  k ^ ) + a3b1 ( k ^   x  i ^ ) + a3b2 ( k ^   x  j ^ )

=   a1b2  k ^    - a1b3  j ^    - a2b1  k ^     + a2b3  i ^    + a3b1  j ^    - a3b2  i ^

â‡’ a â†’   x b â†’   = ( a2b3  - a3b2)  i ^    - (a1b3 -  a3b1) j ^    + (a1b2  - a2b1)  k ^

The cross product of two vectors can also be expressed as a determinant.

â‡’  a â†’   x b â†’   =  i ^ j ^ k ^ a 1 a 2 a 3 b 1 b 2 b 3